The Korn inequality for Jones domains

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

THE KORN INEQUALITY FOR JONES DOMAINS

RICARDO G. DUR ´AN, MARIA AMELIA MUSCHIETTI

Abstract. In this paper we prove the Korn inequality, and its generalization toLp, 1< p <, for bounded domains ΩRn,n2, satisfying an (ǫ, δ) condition.

1. Introduction

Since the pioneering work of Korn [12, 13] on linear elasticity equations, the inequality named after him, in its different forms, has been the subject of a great number of papers. An interesting review article, where connections with other inequalities and several applications are described, was written by Horgan [7].

Given an open domain Ω⊂Rn,n2, the Korn inequality states that

k∇vkL2(Ω)n×n≤Ckε(v)kL2(Ω)n×n (1.1)

whereε(v) denotes the symmetric part of∇v, namely,

εij(v) = 1 2

∂vi

∂xj +∂vj

∂xi

Of course, (1.1) can not be true for arbitrary functionsvH1(Ω)n since there are functions such that the right hand side vanishes while the left one does not (the so called infinitesimal rigid motions). So, in order to prove the inequality, Korn considered two cases.

The so called first case is to considerv H01(Ω)n. In this case the proof of the

inequality (1.1) is simple and was first given by Korn. Moreover, it can be shown that it holds for any open set Ω (not necessarily bounded) and that the constant can be taken asC=√2 (see for example [10]).

The situation is quite different in the second case, where nowvH1(Ω)nsatisfies

Z

Ω ∂vi

∂xj −

∂vj

∂xi

= 0, for i, j= 1, . . . , n.

This case is fundamental for the analysis of the elasticity equations with traction boundary conditions. In this case, the first correct proofs are likely those given by

2000Mathematics Subject Classification. 74B05. Key words and phrases. Korn inequality; Jones domains.

c

2004 Texas State University - San Marcos.

Submitted March 1, 2004. Published October 27, 2004.

The first author was supported by grant PICT 03-05009 from ANPCyT, by grant PIP 0660/98 from CONICET, and by Fundaci´on Antorchas

The second author was supported by grant 11/X228 from Universidad Nacional de La Plata and by grant PIP 4723/96 from CONICET.

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Friedrichs, in [4] for n= 2 and [5] forn= 2 or 3. Indeed, it is not clear whether the original proof of Korn is correct. We have not checked that proof but we can mention that Friedrichs claimed that he has been unable to verify Korn’s proof for the second case (see the footnote 3 in page 443 of [5]). Also Nitsche mentioned that the original proof of Korn is doubtful (see the introduction of [14]).

A different way of stating the Korn inequality is to say that, for anyvH1(Ω)n,

kvkH1(Ω)n≤C{kvkL2(Ω)n+kε(v)kL2(Ω)n×n} (1.2)

Indeed, (1.1) in the second case, can be derived from (1.2) by using compactness arguments (see for example [10]), provided that H1(Ω) is compactly imbedded in

L2(Ω), which is true under rather general assumptions on Ω. One the other hand,

if (1.1) holds in the second case, then (1.2) also holds (see for example [2]). An important difference between the first and second case is that, in the last one, (1.1) does not hold for arbitrary domains Ω. Indeed, it is known that the Korn inequality in the second case is not true when the domain has external cusps. The papers [6, 18] presented counter-examples showing this fact. Moreover, in the old paper [4], Friedrichs gave a very nice counter-example for an inequality which can be derived from (1.1). Suppose that Ω is a two dimensional domain and that

f(z) =φ(x, y) +iψ(x, y)

is an analytic function of the variablez=x+iyin Ω withR

Ωφ= 0. Then, Friedrichs

proved in [4], that under suitable assumptions on Ω, there exists a constant C

depending only on Ω, such that

kL2(Ω)≤CkψkL2(Ω) (1.3)

and he showed that this estimate is not true for some domains which have external cusps (see [4, page 343]). On the other hand, it is not difficult to see that the second case of Korn inequality implies (1.3) whenever the domain Ω is simply connected. This was proved by Horgan and Payne in [8] (they assume that the boundary is smooth but this is not needed for this implication).

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An interesting question is whether the Korn inequality in the second case holds for domains more general than Lipschitz. In his paper [9], Jones introduced the notion of (ǫ, δ) domain and prove that the extension theorem in Sobolev spaces holds if and only if the domain is of this class (which is much more general than the Lipschitz class).

As we said above, some of the arguments to prove the Korn inequality are re-lated to those used for the extension theorem. This fact, together with the above mentioned counter-examples which resemble those for the extension theorem, give rise to the following question:

Is the Korn inequality in the second case (or its equivalent form (1.2)) valid for bounded (ǫ, δ) domains?

In this paper we prove that the answer to this question is positive. In order to do that, we modify the arguments of Jones [9], to construct an extension preserving the norm on the right hand side of (1.2). The key point in our construction is the use of the inequality (1.1), in the second case, on cubes or finite union of cubes. Once we have the extension, the argument concludes by applying (1.2) on a ball containing our original domain.

Our arguments apply also for the generalization of the Korn inequality to Lp, 1< p <.

2. Proof of the Korn inequality

Let ΩRn,n2, be a bounded open domain satisfying the (ǫ, δ) condition of Jones, namely, for any x, y Ω such that|xy| < δ there is a rectifiable arc Γ joiningxtoy and satisfying

l(Γ) 1

ǫ|x−y|,

d(z)≥ ǫ|x−z||y−z|

|xy| ∀z∈Γ

wherel(Γ) denotes the arclength of Γ andd(z) is the distance fromz to Ωc. For 1 < p <, we will prove that there exists a constant C =C(Ω, p, ǫ, δ, n) such that, for anyvW1,p(Ω)n,

kvkW1,p(Ω)n≤C{kvkLp(Ω)n+kε(v)kLp(Ω)n×n} (2.1)

As mentioned in the introduction, this inequality is equivalent to

k∇vkLp(Ω)n×n≤Ckε(v)kLp(Ω)n×n (2.2)

for anyvW1,p(Ω)n satisfying

Z

Ω ∂vi

∂xj −

∂vj

∂xi

= 0, for i, j= 1, . . . , n.

which, forp= 2, is known as “the second case” of Korn inequality.

Remark 2.1. It is easy to see, by a simple scaling argument, that the constant in the inequality (2.2) depends only on the shape of the domain Ω.

Through the rest of the paper we will use the letterC to denote different con-stants which may depend on ǫ, δ, n, pand the diameter of Ω (we will indicate this dependence only some times for the sake of clarity).

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norm on the right hand side of (2.1) and such thatEvhas compact support. We will construct such an extension by modifying appropriately the extension operator of Jones [9]. Since, for (ǫ, δ) domains, smooth functions are dense inW1,p(Ω) (see [9]), it is enough to prove inequality (2.1) for vW1,∞(Ω)n and so, we will construct the extension for suchv.

Given S Rn, a measurable set with|S|>0, we callx its barycenter and for

vW1,∞(Ω)n we associate withS andv the affine vector field

PS(v)(x) =a+B(x−x), (2.3)

whereaRn andB = (bij)Rn×n are defined by

a= 1

|S|

Z

S

v and bij = 1 2|S|

Z

S

∂vi

∂xj −

∂vj

∂xi

. (2.4)

Observe that, since bij = −bji, PS(v) is an “infinitesimal rigid motion”, i.e., it satisfies

ε(PS(v)) = 0. (2.5)

Moreover,aandB have been chosen in such a way that

Z

S

∂(vi−PS(v)i)

∂xj −

∂(vj−PS(v)j)

∂xi

= 0, (2.6)

Z

S

(vPS(v))) = 0. (2.7)

Assume now that the inequality (2.2) holds inS. Then, in view of (2.5) and (2.6) we have

k∇(vPS(v)))kLp(S)n×n≤Ckε(v)kLp(S)n×n (2.8)

where the constantC depends onpand on the shape (but not on the scale!) ofS

(see Remark 2.1).

Now, from (2.7) and the Friedrichs-Poincar´e inequality for functions with van-ishing mean value and denoting withd(S) the diameter ofS, we have

kvPS(v)kLp(S)n≤Cd(S)k∇(v−PS(v))kLp(S)n×n (2.9)

and therefore,

kvPS(v)kLp(S)n≤Cd(S)kε(v)kLp(S)n×n (2.10)

where, again, the constant C depends only on p and on the shape ofS. On the other hand, it is easy to see that

k∇PS(v)kL∞(S)n×n≤ k∇vkL(S)n×n (2.11)

and so,

k∇(vPS(v))kL∞(S)n×n≤2k∇vkL(S)n×n (2.12)

and therefore, using (2.9) withp=∞, we obtain

kvPS(v)kL∞(S)n≤Cd(S)k∇vkL(S)n×n (2.13)

where, also here, the constantC depends only on the shape ofS.

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Recall that any open set Ω⊂Rn admits a Whitney decomposition into closed dyadic cubes Sk (see [19, 16]) , i.e., Ω = ∪kSk, such that, if ℓ(S) denotes the edgelength of a cubeS,

1≤ dist((SSk, ∂Ω)

k) ≤

4√n k , (2.14)

Sj0S0k=∅ ifj6=k , (2.15) 1

4 ≤

ℓ(Sj)

ℓ(Sk)≤

4 ifSj∩Sk 6=∅. (2.16)

Let W1 ={Sk} be a Whitney decomposition of Ω and W2 ={Qj} one of (Ωc)0. We define

W3=

Qj ∈W2 : ℓ(Qj)≤

ǫδ

16n

It was shown by Jones (see Lemmas 2.4 and 2.8 of [9]) that, for eachQj ∈W3, it

is possible to choose a “reflected” cubeQ∗j =Sk ∈W1 such that

1 ℓ(Sk)

ℓ(Qj) ≤4 and d(Qj, Sk)≤Cℓ(Qj)

and moreover, if Qj, Qk ∈ W3 and Qj∩Qk 6=∅, there is a chain Fj,k = {Q∗j =

S1, S2,· · ·, Sm=Q∗k}(i.e,Sj∩Sj+16=∅) of cubes inW1, connectingQ∗j toQ∗k and withmC(ǫ, δ).

It is known that, associated with a Whitney decomposition, there exists a par-tition of unity{φj} such thatφj∈C∞(Rn),supp φj⊂ 1716Qj, 0≤φj ≤1,

X

Qj∈W3

φj≡1 on

[

Qj∈W3 Qj

and

|∇φj| ≤Cℓ(Qj)−1 ∀j (see [16, 19]). Now, given vW1,∞(Ω)n, let P

j =PQ∗

j(v) defined as in (2.3) and

(2.4) withS=Q∗

j. Then, we defineEv, the extension toRn ofv, in the following way,

Ev= X Qj∈W3

Pjφj in (Ωc)0,

Ev=v in Ω.

Since|∂Ω|= 0 (see Lemma 2.3 in [9]), it follows thatEv is definedp.p. inRn. The arguments of the following lemmas are similar to those in [9]. In particular we will make repeated use of Lemma 2.1 of [9] which says,

Lemma 2.2. Let Q be a cube and F, GQbe two measurable subsets such that |F|,|G| ≥γ|Q| for someγ >0. IfP is a polynomial of degree1 then,

kPkLp(F)≤C(γ)kPkLp(G).

Lemma 2.3. Let F ={S1,· · ·, Sm} be a chain of cubes inW1. Then,

kPS1(v)−PSm(v)kLp(S1)n ≤Cc(m)ℓ(S1)kε(v)kLp(∪jSj)n×n, (2.17) and

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Proof. We will use (2.10) withS being a cube or a union of two neighboring cubes. In view of (2.16) there are a finite number, depending only on the dimension n, of possible shapes for the union of two neighboring cubes, and so, we can take a uniform constant in (2.10).

Using Lemma 2.2 we have

kPS1(v)−PSm(v)kLp(S1)n

m−1 X

r=1

kPSr(v)−PSr+1(v)kLp(S1)n

≤c(m) m−1

X

r=1

kPSr(v)−PSr+1(v)kLp(Sr)n

≤c(m) m−1

X

r=1

{kPSr(v)−PSr∪Sr+1(v)kLp(Sr)n

+kPSr∪Sr+1(v)−PSr+1(v)kLp(Sr+1)n}

≤c(m) m−1

X

r=1

{kvPSr(v)kLp(Sr)n+kv−PSr+1(v)kLp(Sr+1)n

+kvPSr∪Sr+1(v)kLp(Sr∪Sr+1)n} ≤Cc(m)ℓ(S1)kε(v)kLp(jSj)n×n

where we have used (2.10). The proof of (2.18) is analogous using now (2.13).

Now, for each Qj, Qk ∈ W3 such that Qj ∩Qk 6= ∅, we choose a chain Fj,k connectingQ∗j to Q∗k and withmC(ǫ, δ) and define

F(Qj) =

[

Qk∈W3, Qj∩Qk6=∅

Fj,k

then,

X

Qk, Qj∩Qk6=∅

χ∪Fj,k

L(Rn)≤C ∀Qj ∈W3 (2.19)

The following lemmas will allow us to control the norms ofEv,ε(Ev) and∇(Ev) in (Ωc)0.

Lemma 2.4. ForQ0∈W3 we have kEvkLp(Q

0)n≤C{kvkLp(Q∗0)n+ℓ(Q0)kε(v)kLp(∪F(Q0))n×n}, (2.20) kε(Ev)kLp(Q0)n×n ≤Ckε(v)kLp(F(Q0))n×n, (2.21) kEvkL∞(Q

0)n≤C{kvkL∞(Q∗0)n+ℓ(Q0)k∇vkL∞(∪F(Q0))n×n}, (2.22) k∇(Ev)kL∞(Q

0)n×n≤Ck∇vkL∞(∪F(Q0))n×n}. (2.23)

Proof. OnQ0we have

Ev= X Qj∈W3

Pjφj

Now, sinceP

Qj∈W3φj≡1 on∪Qj∈W3Qj, then

k X

Qj∈W3

PjφjkLp(Q

0)n≤ kP0kLp(Q0)n+k

X

Qj∈W3

(Pj−P0)φjkLp(Q

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Using Lemma 2.2 and (2.10), we have

I=kP0kLp(Q

0)n≤CkP0kLp(Q∗0)n ≤CkP0−vkLp(Q

0)n+kvkLp(Q ∗ 0)n ≤C{kvkLp(Q

0)n+ℓ(Q0)kε(v)kLp(Q ∗ 0)n×n}

Now, since onQ0there are a finite number (depending only onn) of non-vanishing

φj and 0≤φj ≤1, to boundII it is enough to boundk(Pj−P0)kLp(Q

0)n. But,

using (2.17) and again Lemma 2.2, we have

k(Pj−P0)kLp(Q

0)n≤Ck(Pj−P0)kLp(Q∗0)n ≤Cℓ(Q0)kε(v)kLp(∪F0,j)n×n

and therefore, summing up and using (2.19) we obtain (2.20). Analogously, we can prove (2.22) using now (2.13) and (2.18).

Now, callingPr

j the components ofPj and recalling thatε(Pj) = 0 we have

εrs(Pjφj) = 1 2P

r j

∂φj

∂xs +1

2P s j

∂φj

∂xr

(2.24)

OnQ0,

Ev=P0+ X

Qj∈W3

(Pj−P0)φj

and therefore, sinceε(P0) = 0 we have,

ε(Ev) = X Qj∈W3

ε((Pj−P0)φj)

but, there are at mostC cubesQj such thatφj does not vanishes inQ0 and these

Qj intersectQ0 and therefore,ℓ(Qj)≥ 14ℓ(Q0). Thus,

|∇φj| ≤Cℓ(Q0)−1

wheneverφj6= 0 for somex∈Q0. Then, for these values ofj, it follows from (2.24)

and (2.17) that

kε((Pj−P0)φj)kLp(Q

0)n×n≤Cℓ(Q0) −1

kPj−P0kLp(Q 0)n ≤Cℓ(Q0)−1kPj−P0kLp(Q

0)n≤Ckε(v)kLp(∪F0,j)n×n

Summing up inj, we obtain (2.21).

The proof of (2.23) is similar to that of (2.21) but using now (2.11). Indeed, we have

∇(Ev) =P0+ X

Qj∈W3

∇((Pj−P0)φj)

and we have to estimate also the termsP0 and∇(Pj−P0). But,

k∇P0kL∞(Q

0)n×n≤Ck∇P0kL∞(Q∗0)n×n ≤Ck∇vkL∞(Q0)n×n,

and

k∇(Pj−P0)kL∞(Q

0)n×n ≤Ck∇(Pj−P0)kL∞(Q∗0)n×n ≤Ck∇(Pj−P0)kL∞(Q

0∪Q ∗ j)n

×n≤Ck∇vkL(Q∗ 0∪Q

∗ j)n

×n

≤Ck∇vkL∞(F j,0)n×n}

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Lemma 2.5. ForQ0∈W2\W3 we have kEvkLp(Q0)n≤C

X

Qj∈W3, Qj∩Q06=∅

{kvkLp(Q

j)n+kε(v)kLp(Q ∗

j)n×n}, (2.25)

kε(Ev)kLp(Q

0)n×n≤C

X

Qj∈W3, Qj∩Q06=∅

{kvkLp(Q

j)n+kε(v)kLp(Q ∗ j)n

×n}, (2.26)

kEvkL∞(Q 0)n≤C

X

Qj∈W3, Qj∩Q06=∅

{kvkL∞(Q

j)n+k∇vkL∞(Q ∗

j)n×n}, (2.27)

k∇(Ev)kL∞(Q

0)n×n≤C

X

Qj∈W3, Qj∩Q06=∅

{kvkL∞(Q

j)n+k∇vkL∞(Q ∗

j)n×n} (2.28)

Proof. Ifφj does not vanish onQ0 thenQj∩Q06=∅and so,

ℓ(Qj)≥ 1

4ℓ(Q0)≥

ǫδ

64n

therefore, onQ0, we have

|Ev|=| X Qj∈W3, Qj∩Q06=∅

φjPj| ≤C

X

Qj∈W3, Qj∩Q06=∅ |Pj|,

but

kPjkLp(Q

0)n≤CkPjkLp(Q0∗)n≤C{kv−PjkLp(Q ∗

0)n+kvkLp(Q ∗ 0)n}.

Now, since Ω is bounded,ℓ(Q∗j) is bounded by a constant depending only on Ω and therefore, using (2.10), we obtain

kPjkLp(Q

0)n≤C{kε(v)kLp(Q∗0)n

×n+kvkLp(Q

0)n} (2.29)

and therefore, (2.25) is proved. On the other hand, onQ0we have

εrs(Ev) =1 2

X

Qj∈W3, Qj∩Q06=∅

{Pjr∂φj ∂xs

+Pjs∂φj ∂xr}

but,ℓ(Qj)≥64ǫδn, and so|∇φj| ≤C and (2.26) follows using again (2.29). Finally, (2.27) and (2.28) are obtained by similar arguments, using now (2.11).

Corollary 2.6. If vW1,∞(Ω) then

kEvkLp((Ωc)0)n+kε(Ev)kLp((Ωc)0)n×n≤C{kvkLp(Ω)n+kε(v)kLp(Ω)n×n} (2.30) and

kEvkW1,∞((Ωc)0)n ≤CkvkW1,∞(Ω)n (2.31) Proof. It follows immediately from Lemmas 2.4 and 2.5 by summing up over all

Q0∈W2.

We can now state the main theorem which follows from the results above and arguments given in [9].

Theorem 2.7. If Ω is a bounded (ǫ, δ) domain and 1 < p < , there exists a constant C depending on ǫ, δ, n, p and the diameter of Ω such that, for any v W1,p(Ω)n,

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Proof. By density, it is enough to prove the inequality for smoothv. So, we assume thatvW1,∞(Ω)n. As we already said, the extensionEvis defined almost every-where inRn because|∂Ω|= 0. Moreover, the support ofEv is contained in a ball

B.

In view of (2.31) and the fact that|∂Ω|= 0, ifEvis a continuous function then, it is Lipschitz. But, the continuity ofEv can be proved exactly as in Lemma 3.5 of [9]. Therefore,EvW1,p(B)n and so, from (2.30), it follows that

kEvkLp(B)n+kε(Ev)kLp(B)n×n ≤C{kvkLp(Ω)n+kε(v)kLp(Ω)n×n} (2.33)

but, using the Korn inequality for smooth domains, we have

kEvkW1,p(B)n ≤C{kEvkLp(B)n+kε(Ev)kLp(B)n×n}

which together with (2.33) concludes the proof.

We conclude the paper stating two consequences of our main theorem which fol-lows by known arguments. The first result is the second case of the Korn inequality and the second one is the Friedrichs inequality for complex analytic functions (and their generalizations toLp).

Corollary 2.8. IfΩRn is a bounded (ǫ, δ)domain and 1< p <, there exists

a constantCdepending onǫ, δ, n, psuch that, for anyvW1,p(Ω)n which satisfies

Z

Ω ∂vi

∂xj −

∂vj

∂xi

= 0,

k∇vkLp(Ω)n×n≤Ckε(v)kLp(Ω)n×n.

Corollary 2.9. If Ω R2 is a bounded simply connected(ǫ, δ) domain and 1 < p <, there exists a constant C depending onǫ, δ, n, p such that for any analytic function of the variablez=x+iy inΩ,f(z) =φ(x, y) +iψ(x, y), withR

Ωφ= 0,

kLp(Ω)≤CkψkLp(Ω).

Conclusions. We have proved that, if Ω is an (ǫ, δ) domain, the Korn inequality holds on Ω. It is interesting to observe that the converse is not true. Indeed, consider the two dimensional domain

Ω = (−1,1)2\ {(x,0) :x[0,1)}.

Since Ω can be written as the union of two Lipschitz domains, the Korn inequality is valid on Ω. On the other hand, it is not difficult to see that Ω is not an (ǫ, δ) domain. Therefore, to give a characterization of the domains satisfying the Korn inequality is an interesting problem which remains open.

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[17] T. W. Ting,Generalized Korn’s inequalities, Tensor25, pp. 295-302, 1972.

[18] N. Weck, Local compactness for linear elasticity in irregular domains, Math. Meth. Appl. Sci.,17, pp. 107-113, 1994.

[19] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc.,36, pp. 63-89, 1934.

Ricardo G. Dur´an

Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina

E-mail address:rduran@dm.uba.ar

Maria Amelia Muschietti

Departamento de Matem´atica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Casilla de Correo 172, 1900 La Plata, Provincia de Buenos Aires, Argentina

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